1. Introduction
In this paper, we deal with the convolutional noise arising at the output from a blind deconvolutional process. A blind deconvolution process arises in many applications such as seismology, underwater acoustic, image restoration and digital communication [1] - [7] . A higher convolutional noise may lead to more errors in the recovering process [8] . According to [9] [10] , the convolutional noise power depends on the step-size parameter, equalizer’s tap length, input signal statistics, channel characteristics, characteristics of the chosen blind equali- zation technique and on the signal to noise ratio (SNR). Thus, those fluctuations of the convolutional noise that are near the mean value are well understood and investigated. By well understood we mean that it is well understood how to control the convolutional noise power (thus also the convolutional noise fluctuation near the mean) via the step-size parameter or equalizer’s tap length for instance. But up to now, those fluctuations of the convolutional noise that are larger than the typical ones (which are near the mean), which occur rarely, were not addressed.
The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory [11] . According to [12] , the theory of large deviations deals with the pro- babilities of rare events (or fluctuations) that are exponentially small as a function of some parameters, e.g., the number of random components of a system, the time over which a stochastic system is observed, the amplitude of the noise perturbing a dynamical system or the temperature of a chemical reaction. The reader may refer also to [13] - [17] for further information on the theory of large deviations.
In this paper we address indirectly those fluctuations of the convolutional noise that are larger than the typical ones, which occur very rare. Namely, we consider the fluctuations of the arithmetic average (sample mean) of the real part of consecutive convolutional noises which deviate from the mean of order higher than the typical fluctuations. As already mentioned, typical fluctuations are those fluctuations that fluctuate near the mean, while the other fluctuations that deviate from the mean of order higher than the typical ones are considered as rare events. Via the large deviation theory, we obtain a closed-form approximated expression for the probability that these rare events may occur as a function of the step-size parameter, equalizers’s tap length, SNR, input signal statistics, characteristics of the chosen equalizer (based on a cost function where the error of the equalized output can be expressed as a polynomial function of order up to three), channel power and the amount of deviation from the mean. Based on this new expression we are able to evaluate approximately the amount of deviation from the mean of those fluctuations considered as rare events as a function of the system’s parameters (step-size parameter, equalizers’s tap length, SNR, input signal statistics, characteristics of the chosen equalizer and channel power), for a pre-given probability that these events may occur.
The paper is organized as follows: after having described the system under consideration in Section 2 we evaluate in Section 3 approximately the amount of deviation from the mean of those fluctuations considered as rare events as a function of the systems parameters (step-size parameter, equalizerss tap length, SNR, input signal statistics, characteristics of the chosen equalizer and channel power), for a pre-given probability that these events may occur. Section 4 is our conclusion.
2. System Description
The system under consideration is illustrated in Figure 1, where we make the following assumptions:
1) The input sequence 
belongs to a two independent quadrature carrier case constellation input with variance 
where 
and 
are the real and imaginary parts of 
respectively.
2) The unknown channel 
is a possibly nonminimum phase linear time-invariant filter in which the transfer function has no “deep zeros”, namely, the zeros lie sufficiently far from the unit circle.
3) The equalizer 
is a tap-delay line.
4) The noise 
is an additive Gaussian white noise with zero mean and variance 
where 
is the expectation operator.
![]()
Figure 1. Block diagram of a baseband communication system.
The transmitted sequence 
is transmitted through the channel 
and is corrupted with noise
. Therefore, the equalizer’s input sequence 
may be written as:
![]()
(1)
where “![]()
” denotes the convolution operation. The equalized output sequence is defined by:
![]()
(2)
where ![]()
is the convolutional noise (convolutional error) due to non-ideal equalizer’s coefficients
![]()
and![]()
. The update mechanism of the equalizer’s coefficients is given by
[18] :
![]()
(3)
where ![]()
is the step-size parameter, ![]()
stands for the conjugate operation and ![]()
is a predefined
cost function that characterizes the intersymbol interference, see [19] - [26] . Minimizing this ![]()
with
respect to the equalizer parameters will reduce the convolutional error. In the following we deal with those
equalization methods where ![]()
can be defined as a polynomial function ![]()
of order up to
three as is in the case of [19] . Thus, in this work we consider the following update mechanism of the equalizer’s coefficients:
![]()
(4)
3. The Deviation from the Mean
In this section we obtain a closed-form approximated expression for the amount of deviation from the mean of those fluctuations considered as rare events as a function of the system’s parameters (step-size parameter, equalizers’s tap length, SNR, input signal statistics, characteristics of the chosen equalizer and channel power), for a pre-given probability that these events may occur.
Theorem 1. For the following assumptions:
1) The convolutional noise![]()
, is a zero mean, white Gaussian process with variance
![]()
, where![]()
. ![]()
is the real part of![]()
.
2) The source signal ![]()
is a complex input where the real part of ![]()
is independent with the imaginary part of![]()
, with known variance ![]()
and higher moments. The mean of the input sequence is
zero. Namely,![]()
.
3) The convolutional noise ![]()
and the source signal are independent.
4) ![]()
can be expressed as a polynomial function of the equalized output namely as ![]()
of
order three.
5) The gain between the source and equalized output signal is equal to one.
6) The convolutional noise ![]()
is independent with![]()
.
The amount of deviation ![]()
from the mean of those fluctuations considered as rare events as a function of the system’s parameters, given the probability that these events occur can be expressed by:
![]()
(5)
where the probability that the fluctuations of the arithmetic average (sample mean) of the real part of consecutive convolutional noises which deviate from the mean of order higher than the typical ones is expressed by:
![]()
(6)
and ![]()
is given according to [27] :
![]()
(7)
![]()
(8)
xr is the real part of![]()
, R is the channel length, N is the equalizer’s tap length, ![]()
and![]()
, ![]()
, ![]()
are properties of the chosen equalizer and found by:
![]()
(9)
where ![]()
is the real part of ![]()
and![]()
, ![]()
are the real and imaginary parts of the equalized output ![]()
respectively.
Proof. Let us first define:
![]()
(10)
Next, by using assumption 1 from this section, we may write:
![]()
(11)
The propability density function (pdf) for ![]()
(10) is
![]()
(12)
since a sum of Gaussian random variables is also exactly Gaussian-distributed. According to [11] , a large deviation approximation is obtained from this exact result by neglecting the term![]()
, which is subdominant with respect to the decaying exponential, thereby obtaining [11]
![]()
(13)
The Central Limit Theory (CLT) governs random fluctuations only near the mean-deviations from the mean
of the order of ![]()
[14] . Fluctuations which are of the order of ![]()
are, relative to typical fluctuations,
much bigger: they are large deviations from the mean [14] . They happen only rarely, and so large deviation theory is often described as the theory of rare events-events which take place away from the mean, out in the tails of the distribution; thus large deviation theory can also be described as a theory which studies the tails of distributions [14] . According to [11] [14] , the probability that
![]()
(14)
where according to [11] , ![]()
is used to stress that as ![]()
the dominant part of ![]()
is
the decaying exponential![]()
. Thus, based on (14), we may write:
![]()
(15)
where
![]()
(16)
and
![]()
(17)
Thus from (17) and (13) we have:
![]()
(18)
Next we turn to find a closed-form approximated expression for![]()
. Based on [27] , the expression for the
convolutional noise power ![]()
for the noisy and non-biased input case ![]()
is given by (7), (8) and (9). This completes our proof.
4. Conclusion
In this paper we dealt with the fluctuations of the arithmetic average (sample mean) of the real part of consecutive convolutional noises which deviate from the mean of order higher than the typical ones. Via the large deviation theory, we obtained a closed-form approximated expression for the amount of deviation from the mean of those fluctuations considered as rare events as a function of the system’s parameters (step-size pa- rameter, equalizerss tap length, SNR, input signal statistics, characteristics of the chosen equalizer and channel power), for a pre-given probability that these events may occur.
Acknowledgements
We thank the Editor and the referee for their comments.